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%
% skalarprodukt.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Skalarprodukt}
\vspace{-15pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.48\textwidth}
\begin{block}{Positiv definite, symmetrische Bilinearform}
$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$
\begin{itemize}
\item
Bilinear:
\begin{align*}
\langle \alpha u+\beta v,w\rangle
&=
\alpha\langle u,w\rangle
+
\beta\langle v,w\rangle
\\
\langle u,\alpha v+\beta w\rangle
&=
\alpha\langle u,v\rangle
+
\beta\langle u,w\rangle
\end{align*}
\item
Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$
\item
$\langle x,x\rangle >0 \quad\forall x\ne 0$
\end{itemize}
\end{block}
\end{column}
\begin{column}{0.48\textwidth}
\begin{block}{Positive definite, hermitesche Sesquilinearform}
$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{C}$
\begin{itemize}
\item
Sesquilinear:
\begin{align*}
\langle \alpha u+\beta v,w\rangle
&=
\overline{\alpha}\langle u,w\rangle
+
\overline{\beta}\langle v,w\rangle
\\
\langle u,\alpha v+\beta w\rangle
&=
\alpha\langle u,v\rangle
+
\beta\langle u,w\rangle
\end{align*}
\item
Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$
\item
$\langle x,x\rangle >0 \quad\forall x\ne 0$
\end{itemize}
\end{block}
\end{column}
\end{columns}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.28\textwidth}
\begin{block}{$2$-Norm}
$\|v\|_2^2 = \langle v,v\rangle$
\\
$\|v\|_2 = \sqrt{\langle v,v\rangle}$
\end{block}
\end{column}
\begin{column}{0.78\textwidth}
\begin{itemize}
\item $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$
\item $\| \lambda v \|_2
=
\sqrt{\langle \lambda v,\lambda v\rangle\mathstrut}
=
\sqrt{\overline{\lambda}\lambda\langle v,v\rangle}
=
|\lambda|\cdot \|v\|_2$
\item
\raisebox{-8pt}{
$\begin{aligned}
\|u+v\|_2^2 &= \|u\|_2^2 + 2{\color{red}\operatorname{Re}\langle u,v\rangle} + \|v\|_2^2
\\
(\|u\|_2+\|v\|_2)^2 &= \|u\|_2^2 + 2{\color{red}\|u\|_2\|v\|_2} + \|v\|_2^2
\end{aligned}$}
\end{itemize}
\end{column}
\end{columns}
\end{frame}
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